Dec 10, 2017 - (Dated: December 10, 2017). From 21 independent Baryon Acoustic Oscillation (BAO) measurements we obtain the following sum of masses of ...
Constraints on neutrino masses from Baryon Acoustic Oscillation measurements B. Hoeneisen1 1
Universidad San Francisco de Quito, Quito, Ecuador (Dated: December 10, 2017)
arXiv:1712.03533v1 [hep-ph] 10 Dec 2017
From 21 independent Baryon Acoustic Oscillation P (BAO) measurements we obtain the following sum of masses of active Dirac or Majorana neutrinos: mν = 0.711−0.335·δh+0.050·δb±0.063 eV, where δh ≡ (h − 0.678)/0.009 and δb ≡ (Ωb h2 − 0.02226)/0.00023. be combined P This result may with independent measurements that constrain the parameters mν , h, and Ωb h2 . For δh = ±1 and δb = ±1, we obtain mν < 0.43 eV at 95% confidence.
We extend the analysis presented in Ref. [1] to include neutrino masses. The present analysis has three steps: (1) we calculate the distance of propagation rs , in units of c/H0 , referred to the present time, of sound waves in the photon-electron-baryon plasma until decoupling by numerical integration of Eqs. (16) and (17) of Ref. [1]; (2) we fit the Friedmann equation of evolution of the universe to 21 independent Baryon Acoustic Oscillation (BAO) distance measurements listed in [1] used as uncalibrated standard rulers and obtain the length d of these rulers, in units of c/H0 , referred to the present time; and (3) we set rs = d
(1)
P to constrain the sum of neutrino masses mν . c is the speed of light, and H0 ≡ 100h km s−1 Mpc−1 is the present day Hubble expansion parameter. The main body of this article assumes: (1) flat space, i.e. Ωk = 0, and (2) constant dark energy density relative to the critical density, i.e. ΩDE independent of the expansion parameter a. These constraints are in agreement with all observations to date [1, 2]. Results without these constraints are presented in the appendix. To be specific we consider three active neutrino flavors with P three eigenstates with nearly the same mass mν , so mν = 3mν . This is a useful scenario to consider since our current limits on m2ν are much larger than the mass-squared-differences ∆m2 and ∆m221 obtained from neutrino oscillations [2]. These neutrinos become non-relativistic at a neutrino temperature Tν = mν /3.15 or a photon temperature T = mν (11/4)1/3 /3.15. The corresponding expansion parameter is aν = T0 /T = 5.28 × 10−4 (1eV /mν ). The matter density relative to the present critical density P is Ωm /a3 for a > aν . Ωm includes the density Ων = h−2 mν /94eV of Dirac or Majorana neutrinos that are non-relativistic today. Note that for Dirac neutrinos we are considering the scenario in which right-handed neutrinos and left-handed anti-neutrinos are sterile and never achieved thermal equilibrium. Our results can be amended for other specific scenarios. For a < aν we take the matter density to be (Ωm − Ων )/a3 . The radiation density is Ωγ Neq /(2a4 ) for a < aν , where Neq = 3.36 for three flavors of Dirac (mostly) left-handed neutrinos and
right-handed anti-neutrinos. We also take Neq = 3.36 for three active flavors of Majorana left-handed and righthanded neutrinos. For a > aν , we take the radiation density to be (Ωγ Neq /2 − aν Ων )/a4 = Ωγ /a4 . The present density of photons relative to the critical density is Ωγ = 2.473 × 10−5 h−2 [2]. The data used to obtain d are 18 independent BAO distance measurements with Sloan Digital Sky Survey (SDSS) data release DR13 galaxies in the redshift range z = 0.1 to 0.7 [1, 3–5], two BAO distance measurements in the Lyman-alpha forest (Lyα) at z = 2.36 (crosscorrelation [6]) and z = 2.34 (autocorrelation [7]), and the Cosmic Microwave Background (CMB) correlation angle θMC = 0.010410 ± 0.000005 [2, 8], used as an uncalibrated standard ruler. These 21 independent BAO measurements are summarized in [1]. As a reference we take h = 0.678 ± 0.009,
Ωb h2 = 0.02226 ± 0.00023 (2)
(at 68% confidence) from “Planck TT + low P + lensing” data (that does not contain BAO information) [2]. Ωb is the present density of baryons relative to the critical density. Due to correlations and non-linearities we obtain our final result (Eq. (9) below) with a global fit. The following equations are included to illustrate the dependencePof rs and d on the cosmological parameters h, Ωb h2 and mν in limited ranges of interest. Integrating the comoving sound speed of the photon-baryon-electron plasma until adec = 1/(1 + zdec ) with zdec = 1089.9 ± 0.4 [2] we obtain 0.24 0.28 (3) rs ≈ 0.0339 × A × Ωm with A ≈ 0.990 + 0.007 · δh − 0.001 · δb + 0.020 · where
X
mν , (4)
δh ≡ (h − 0.678)/0.009,
(5)
δb ≡ (Ωb h2 − 0.02226)/0.00023.
(6)
To obtain d we minimize the χ2 with 21 terms, corresponding to the 21 BAO observables, with respect to
2 with χ2 /d.f. = 19.7/18.
ΩDE and d, and obtain ODE = 0.718 ± 0.003 and d ≈ 0.0340 ± 0.0002,
(7)
with χ2 per degree of freedom 19.8/19, and correlation coefficient 0.989 (this high correlation coefficient is due to the high precision of θMC ). Setting rs = d we obtain X
mν ≈ 0.73 − 0.35 · δh + 0.05 · δb ± 0.15 eV.
Freeing Ωk and letting ΩDE (a) = ΩDE ·{1+wa ·(1−a)} we obtain Ωk = −0.008 ± 0.004, ΩDE + 2.2Ωk = 0.718 ± 0.004, wa = 0.227 ± 0.069, and
(8)
A more precise result is obtained with a global P fit by minimizing the χ2 with 21 terms varying ΩDE and mν directly. We obtain ΩDE = 0.7175 ± 0.0023 and X
mν = 0.711 − 0.335 · δh + 0.050 · δb ± 0.063 eV, (9)
with χ2 /d.f. = 19.9/19, and correlation coefficient 0.924. This is our main result. Equation (9) is obtained from BAO measurements alone, and is written in a way that can be combined with independent constraints on the P cosmological parameters mν , h and Ωb h2 , such as measurements of the power spectrum of density fluctuations P (k), the CMB, and direct measurements of the Hubble parameter. Setting δh = ±1 and δb = ±1 we obtain the following upper bound on the mass of active neutrinos P mν = 31 mν : mν < 0.43 eV at 95% confidence.
(10)
Appendix
Freeing Ωk and keeping ΩDE constant we obtain Ωk = −0.003 ± 0.006, ΩDE + 2.2Ωk = 0.719 ± 0.003, and X
mν = 0.623 − 0.334 · δh + 0.050 · δb ± 0.191 eV, (11)
with χ2 /d.f. = 19.6/18. Fixing Ωk = 0 and letting ΩDE (a) = ΩDE ·{1+wa ·(1− a)} we obtain ΩDE = 0.716 ± 0.004, wa = 0.064 ± 0.148, and X mν = 0.603 − 0.349 · δh + 0.052 · δb ± 0.257 eV, (12)
0
